Abstract
In a symmetry-regular bar-and-joint framework of given point-group symmetry, all bars and joints occupy general positions with respect to the symmetry elements. The symmetry-extended form of Maxwell’s Rule is applied to this simplest type of framework and is used to derive counts within irreducible representations for infinitesimal mechanisms and states of self stress. In particular, conditions are given for symmetry-regular frameworks to have at least one infinitesimal mechanism (respectively, state of self stress) within each irreducible representation of the point group of the framework. Similar conditions are found for symmetry-regular body-and-joint frameworks.
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Notes
- 1.
Many published character tables for \(\mathcal{S}_{8}\) correctly assign Γ(x, y) to E 1 but also incorrectly assign Γ(R x , R y ) to E 1 instead of to E 3 [28]. The problem extends to some tables for \(\mathcal{S}_{12}\), \(\mathcal{S}_{16}\) and \(\mathcal{S}_{20}\) [1], where the correct assignment is in fact \(\varGamma (R_{x},R_{y}) = E_{(n/2-1)}\).
References
Altmann, S.L., Herzig, P.: Point-Group Theory Tables. Clarendon Press, Oxford (1994)
Atkins, P.W., Child, M.S., Phillips, C.S.G.: Tables for Group Theory. Oxford University Press, London (1970)
Bishop, D.M.: Group Theory and Chemistry. Clarendon Press, Oxford (1973)
Bottema, O.: Die Bahnkurven eines merkwürdigen Zwölfstabgetriebes. Österr. Ingen. Archiv 14, 218–222 (1960)
Connelly, R., Fowler, P.W., Guest, S.D., Schulze, B., Whiteley, W.J.: When is a symmetric pin-jointed framework isostatic? Int. J. Solids Struct. 46, 762–773 (2009)
Fowler, P.W., Guest, S.D.: A symmetry extension of Maxwell’s rule for rigidity of frames. Int. J. Solids Struct. 37, 1793–1804 (2000)
Fowler, P.W., Guest, S.D.: Symmetry analysis of the double banana and related indeterminate structure. In: Drew, H.R., Pellegrino, S. (eds.) New Approaches to Structural Mechanics, Shells and Biological Structures, pp. 91–100. Kluwer Academic, Dordrecht (2002)
Fowler, P.W., Guest, S.D.: Symmetry and states of self stress in triangulated toroidal frames. Int. J. Solids Struct. 39(17), 4385–4393 (2002)
Fowler, P.W., Guest, S.D.: A symmetry analysis of mechanisms in rotating rings of tetrahedra. Proc. R. Soc.: Math. Phys. Eng. Sci. 461, 1829–1846 (2005)
Fowler, P.W., Guest, S.D., Tarnai, T.: A symmetry treatment of Danzerian rigidity for circle packing. Proc. R. Soc. A: Math. Phys. Eng. Sci. 464, 3237–3254 (2008)
Guest, S.D.: Tensegrities and rotating rings of tetrahedra: a symmetry viewpoint of structural mechanics. Philos. Trans. R. Soc. Lond. 358(358), 229–243 (2000)
Guest, S.D., Fowler, P.W.: A symmetry-extended mobility rule. Mech. Mach. Theory 40, 1002–1014 (2005)
Guest, S.D., Fowler, P.W.: Symmetry conditions and finite mechanisms. J. Mech. Mater. Struct. 2(2), 293–301 (2007)
Guest, S.D., Fowler, P.W.: Mobility of ‘N-loops’: bodies cyclically connected by intersecting revolute hinges. Proc. R. Soc. A: Math. Phys. Eng. Sci. 466, 63–77 (2010)
Hunt, K.H.: Kinematic Geometry of Mechanisms. Clarendon Press, Oxford (1978)
Jahn, H., Teller, E.: Stability of polyatomic molecules in degenerate electronic states. I. Orbital degeneracy. Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 161(905), 220–235 (1937)
James, G.D., Liebeck, M.W.: Representations and Characters of Groups. Cambridge University Press, Cambridge/New York (2001)
Kangwai, R.D., Guest, S.D.: Detection of finite mechanisms in symmetric structures. Int. J. Solids Struct. 36(36), 5507–5527 (1999)
Kangwai, R.D., Guest, S.D.: Symmetry-adapted equilibrium matrices. Int. J. Solids Struct. 37, 1525–1548 (2000)
Kangwai, R.D., Guest, S.D., Pellegrino, S.: An introduction to the analysis of symmetric structures. Comput. Struct. 71, 671–688 (1999)
Kovács, F., Tarnai, T., Guest, S.D., Fowler, P.W.: Double-link expandohedra: a mechanical model for expansion of a virus. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460(2051), 3191–3202 (2004)
Maxwell, J.C.: On the calculation of the equilibrium and stiffness of frames. Philos. Magazine. 27, 294–299 (1864)
Pellegrino, S., Calladine, C.R.: Matrix analysis of statically and kinematically indeterminate frameworks. Int. J. Solids Struct. 22, 409–428 (1986)
Quinn, C.M., McKiernan, J.G., Redmond, D.B.: Mollweide projections and symmetries on the spherical shell. J. Chem. Educ. 61, 569–571, 572–579 (1984)
Schulze, B.: Block-diagonalized rigidity matrices of symmetric frameworks and applications. Contrib. Algebra Geom. 51(2), 427–466 (2010)
Schulze, B.: Symmetry as a sufficient condition for a finite flex. SIAM J. Discret. Math. 24(4), 1291–1312 (2010)
Schulze, B., Whiteley, W.: The orbit rigidity matrix of a symmetric framework. Discret. Comput. Geom. 46(3), 561–598 (2011)
Shirts, R.B.: Correcting two long-standing errors in point group symmetry character tables. J. Chem. Educ. 84(11), 1882–1884 (2007)
Acknowledgements
This work was begun during the workshop on Rigidity and Symmetry, held at the Fields Institute, Toronto, October 17–21, 2011, and gained from discussions there with Prof. Stephen Power (Lancaster). PWF thanks the Fields Institute for financial support of his visit.
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Fowler, P.W., Guest, S.D., Schulze, B. (2014). Mobility in Symmetry-Regular Bar-and-Joint Frameworks. In: Connelly, R., Ivić Weiss, A., Whiteley, W. (eds) Rigidity and Symmetry. Fields Institute Communications, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0781-6_7
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